The PS5 is capable of outputting video at multiple different resolutions. 4k gaming is something that will be supported by a lot of PS5 games but not them all. It is also not a standard for the console. Since the PS5 also supports PS4 games, which mostly run at 1080p, there will be no problem using a PS5 with a 1080p TV.
Even though it will be possible to use a PS5 with a 1080p tv, you should start saving if possible to purchase a new TV that supports 4k resolution with HDR. This is going to be something that more and more games support. IF you want to get the most out of your PS5, this will be something that you will want to have on your TV.